# Dividend Discount Model (DDM)

A security with a greater risk must potentially pay a greater rate of return to induce investors to buy the security. The **required rate of return** (aka **capitalization rate**) is the rate of return required by investors to compensate them for the risk of owning the security.

This capitalization rate can be used to price a stock as the sum of its present values of its future cash flows in the same way that interest rates are used to price bonds in terms of its cash flows. The price of a bond is the sum of the present value of its future interest payments discounted by the market interest rate. Similarly, the **dividend discount model** (aka **DDM**, **dividend valuation model**, **DVM**) prices a stock by the sum of its future cash flows discounted by the required rate of return that an investor demands for the risk of owning the stock. This risk can be determined by the capital asset pricing model. Riskier investments require a higher rate of return to compensate the investors for the risk.

Future cash flows include dividends and the sale price of the stock when it is sold. This DDM price is the **intrinsic value** of the stock. If the stock pays no dividend, then the expected future cash flow is the sale price of the stock.

Intrinsic Value = Sum of Present Value of Future Cash Flows

Intrinsic Value = Sum of Present Value of Dividends + Present Value of Stock Sale Price

Stock Intrinsic Value | = | D_{1}(1+k) ^{1} | + | D_{2}(1+k) ^{2} | + ... + | D_{n}(1+k) ^{n} | + | P (1+k) ^{n} |

Present Value of Stock Sale Proceeds |

In the above equation, it is assumed that 1 dividend is paid at the end of each year and that the stock is sold at the end of the n^{th} year. This is done so that the capitalization rate (k) is an annual rate, since most rates of return are presented as annual rates, which simplifies the discussion. We are only interested in the equation's pedagogical value rather than specific results. Note also that this equation is similar to the formula for calculating bond prices in terms of the yield to maturity, where the dividend payment is replaced with the coupon payment, the stock price is replaced by the par value of the bond, and the capitalization rate is replaced with the yield to maturity to yield the bond price.

Note that if the stock is never sold, then it is essentially a perpetuity, and its price is equal to the sum of the present value of its dividends. Since the DDM considers the current sale price of the stock to be equal to its future cash flows, then it must also be true that the future sale price of the stock is equal to the sum of the cash flows subsequent to the sale discounted by the capitalization rate.

In an efficient market, the market price of a stock is considered equal to the intrinsic value of the stock, where the capitalization rate is equal to the **market capitalization rate**, the average capitalization rate of all market participants.

There are 3 models used in the dividend discount model:

**zero-growth**, which assumes that all dividends paid by a stock remain the same;- the
**constant-growth model**, which assumes that dividends grow by a specific percent annually; - and the
**variable-growth model**, which typically divides growth into 3 phases: a fast initial phase, then a slower transition phase that ultimately ends with a lower rate that is sustainable over a long period.

## Zero-Growth Rate DDM

Since the zero-growth model assumes that the dividend always stays the same, the stock price would be equal to the annual dividends divided by the required rate of return.

Stock's Intrinsic Value = Annual Dividends / Required Rate of Return

This is basically the same formula used to calculate the value of a perpetuity, which is a bond that never matures, and can be used to price **preferred stock**, which pays a dividend that is a specified percentage of its par value. A stock based on the zero-growth model can still change in price if the capitalization rate changes, as it will if perceived risk changes, for instance.

### Example—Intrinsic Value of Preferred Stock

If a preferred share of stock pays dividends of $1.80 per year, and the required rate of return for the stock is 8%, then what is its intrinsic value?

**Intrinsic Value of Preferred Stock** = $1.80/0.08 = **$22.50**.

## Constant-Growth Rate DDM (aka Gordon Growth Model)

The **constant-growth DDM** (aka **Gordon Growth model**, because it was popularized by Myron J. Gordon) assumes that dividends grow by a specific percentage each year, and is usually denoted as *g*, and the capitalization rate is denoted by *k*.

Intrinsic Value | = | D_{1}k - g |

D_{1} = Next Year's Dividendk = Capitalization Rate g = Dividend Growth Rate |

The constant-growth model is often used to value stocks of mature companies that have increased the dividend steadily over the years. Although the annual increase is not always the same, the constant-growth model can be used to approximate an intrinsic value of the stock using the average of the dividend growth and projecting that average to future dividend increases.

Note that if both the capitalization rate and dividend growth rate remains the same every year, then the denominator doesn't change, so the stock's intrinsic value will increase annually by the percentage of the dividend increase. In other words, both the stock price and the dividend amount will increase by the constant-growth factor, *g*.

### Example—Calculating Next Year's Stock Price Using the Constant-Growth DDM

If a stock pays a $4 dividend this year, and the dividend has been growing 6% annually, then what will be the price of the stock next year, assuming a required rate of return of 12%?

**Next Year's Stock Price** = $4 × 1.06 / (12% – 6%) = 4.24 / 0.06 = **$70.67**

**This Year's Stock Price** = $4 / 0.06 = **$66.67**

Growth Rate of Stock Price = $70.67 / $66.67 = 1.06 = Dividend Growth Rate

Note that both the zero-growth rate and the constant-growth rate dividend discount models both value stocks in terms of the dividends they pay and not on any capital gains in the stock price; the **holding period** for the stock is irrelevant; therefore the **holding period return** is equal either to the dividend rate of the zero-growth model or the constant-growth rate.

### Discounted Cash Flow Formula

From the constant-growth dividend discount model, we can infer the market capitalization rate, k, or the rate of return demanded by investors. Note that:

Expected Return = Dividend Yield + Capital Gains Yield

If a stock is held for 1 year, and is bought and sold for its intrinsic value, then the following **discounted cash flow formula** calculates the market capitalization rate:

Capitalization Rate (k) | = | Dividend Yield | + | Capital Gains Yield |

= | D_{1}P _{0} | + | P_{1} – P_{0}P _{0} | |

= | D_{1}P _{0} | + | P_{0}(1+g) – P_{0}P _{0} | |

= | D_{1}P _{0} | + | g | |

g = Dividend Growth Rate |

Often, this is how rates are determined for public utilities by the agencies responsible for setting public rates. Public utilities are generally allowed to charge rates that cover their costs plus a fair market return, with the fair market return being the market capitalization rate.

### Implied Growth Rate and Return on Equity

The constant-growth rate DDM formula can also be algebraically transformed, by setting the intrinsic value equal to the current stock price, to calculate the **implied growth rate**, then using the result, divided by the earnings retention rate, to calculate the **implied return on equity**.

Implied Growth Rate (g) | = | k | – | D_{1}P |

D_{1} = Next Year's Dividendk = Capitalization Rate P = Current Stock Price |

Implied Return on Equity | = | Implied Growth Rate Earnings Retention Rate |

### Example—Calculating the Implied Growth Rate and Return on Equity

If:

- Current Stock Price = $65
- Next Year's Dividend = $4
- Capitalization Rate = 12%
- Earnings Retention Rate = 50%

Then

- Implied Growth Rate = .12 – 4/65 ≈ 5.8%
**Implied Return on Equity**= .058/.5 =**11.6%**

## Variable-Growth Rate DDM

**Variable-growth rate models** (aka **multi-stage growth models**) can take many forms, even assuming the growth rate is different for every year. However, the most common form is one that assumes 3 different rates of growth: an initial high rate of growth, a transition to slower growth, and lastly, a sustainable, steady rate of growth. Basically, the constant-growth rate model is extended, with each phase of growth calculated using the constant-growth method, but using 3 different growth rates of the 3 phrases. The present values of each stage are added together to derive the intrinsic value of the stock.

Sometimes, even the capitalization rate, or the required rate of return, may be varied if changes in the rate are projected.

## Conclusion

The dividend discount model is a useful heuristic model that relates the present stock price to the present value of its future cash flows in the same way that a bond is priced in terms of its future cash flows. However, bond pricing is more exact, especially if the bond is held to maturity, since its cash flows and the interest rate of those cash flows are known with certainty, unless the bond issuer defaults. The dividend discount model, however, depends on projections about company growth rate and future capitalization rates of the remaining cash flows. For instance, in a bear market, the capitalization rate will be higher than in a bull market — investors will demand a higher required rate of return to compensate them for a perceived greater amount of risk. Getting either the capitalization rate or the growth rate wrong will yield an incorrect intrinsic value for the stock, especially since even small changes in either of these factors will greatly affect the calculated intrinsic value. Furthermore, the longer the time considered, the more likely both factors will be wrong. Hence, the true intrinsic value of a stock is unknowable, and, thus, it cannot be determined whether a stock is undervalued or overvalued based on a calculated intrinsic value, since different investors will have a different opinion about the company's future.